Let $J$ be an ideal of $R$, and $M$ a right $R$-module.
Since $Jr \subseteq J$, $M / MJ$ is naturally a right $R$-module. Since it seems relevant to another problem, I am trying to determine $\operatorname{ann}_{R}(M/MJ)$. Clearly, $\operatorname{ann}_{R}(M/MJ) \supseteq \operatorname{ann}_R(M) + J$.
Am I missing anything?
It seems conceivable that there could be an element $r \in R \setminus (\operatorname{ann}_R(M) + J)$ so that $Mr \subseteq MJ$. However, I can't think of a counter example, nor prove that this never happens. I would appreciate a nudge in the right direction.
Edit: I found an easier way to solve the original problem, but I'm still curious about this.
(If this is the question) $\operatorname{ann}(M/JM)$ can differ from $\operatorname{ann}(M)+J$ as it is shown by the following simple example: $R=\mathbb Z$, $M=\mathbb Q$, $J=2\mathbb Z$.